System and method for fast compression of OFDM channel state information (CSI) based on constant frequency sinusoidal approximation

ABSTRACT

A system and method for the efficient compression of the Channel State Information (CSI) in a wireless network with very low complexity and implementation cost. In accordance with the present invention, the CSI can be approximated as the summation of very few sinusoids on constant frequencies and the parameters of the sinusoids can be found efficiently by very simple calculations such as dot products of vectors which are implementable in hardware at very low cost.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims priority to currently U.S. Provisional PatentApplication No. 62/295,871 filed on Feb. 16, 2016 and entitled, “FastCompression of the OFDM Channel State Information (CSI) Based onConstant Frequency Sinusoidal Approximation”.

BACKGROUND OF INVENTION

Today, the demand on higher speed of Wi-Fi is driven by the need tosupport more applications at higher quality. The Wi-Fi protocol has beenconstantly updated to meet such demands with newer technologies, such asadopting Orthogonal Frequency Division Multiplexing (OFDM) since802.11a, Multiple-Input-Multiple-Output (MIMO) since 802.11n, andMulti-User MIMO (MU-MIMO) with the latest 802.11ac. One of the majorbarriers to fully exploiting new technologies such as MU-MIMO, orbeamforming with single user MIMO, is the high overhead of the ChannelState Information (CSI) feedback. The CSI for an antenna pair is avector of complex numbers representing the channel coefficients of theOFDM subcarriers, and is the key to calculating the modulationparameters for the data transmission. In Wi-Fi, the CSI is typicallymeasured at the receiver and is transmitted back to the sender, whichrequires significant overhead. For example, the full CSI for a singleantenna pair on a 20 MHz channel has 64 complex numbers; if there are 9antenna pairs, the full CSI has 9 vectors with 576 complex numbers whichmay exceed 1000 bytes.

The Wi-Fi standard defines options to compress the CSI, such as reducingthe quantization accuracy or the number of subcarriers in the feedback,or using the Given's rotation on the V matrix after the Singular ValueDecomposition (SVD) of the CSI matrix. However, these methods eitherreduce the accuracy of the CSI, or only achieve modest compressionratios. For example, a 3 by 3 complex V matrix can only be compressedinto 6 real numbers, at a compression ratio of 3. The need for CSIfeedback reduction in wireless networks in general is well known in theart.

As such, while there have been many attempts to compress the CSI toreduce the overhead, such as reducing the quantization accuracy or thenumber of subcarriers in the feedback, or using the sinusoidalrepresentation of the CSI, the existing solutions either sacrificeaccuracy or exhibit high computational complexity.

Accordingly, what is needed in the art is an improved system and methodfor compressing the CSI for OFDM that is accurate and computationallyeasy to implement.

SUMMARY OF INVENTION

In various embodiments, the present invention provides a system andmethod, referred to as CSIApx, which is a very fast and lightweightmethod to compress the Channel State Information (CSI) of Wi-Finetworks. In various embodiments, CSIApx approximates the CSI vector asthe linear combination of a small number of base sinusoids on constantfrequencies and uses the complex coefficients of the base sinusoids asthe compressed CSI. While it is well-known that the CSI vector can berepresented as the linear combination of sinusoids, fixing thefrequencies of the base sinusoids is the key novelty of CSIApx, which isguided by the mathematical finding that almost any sinusoid can beapproximated by the same set of base sinusoids with small bounded error.CSIApx enjoys very low computation complexity, because the key steps inthe compression can be pre-computed. Extensive testing of CSIApx withboth experimental and synthesized Wi-Fi channel data confirms thatCSIApx can achieve very good compression ratio with little loss ofaccuracy.

In one embodiment, the present invention provides, a method forcompressing channel state information (CSI) of a wireless channel, whichincludes, receiving, at a receiver of a wireless communication system,an orthogonal frequency division multiplexing (OFDM) wireless signalover a wireless channel and measuring at a receiver of a wirelesscommunication system comprising one or more antenna pair, a channelstate information (CSI) vector of the wireless channel from the receivedOFDM wireless signal for each antenna pair, wherein the CSI vector is anN by 1 vector of complex numbers and wherein each complex numberrepresents an amplitude and a phase of one of N orthogonal frequencydivision multiplexing (OFDM) subcarriers of the wireless channel. Themethod further includes, accessing a plurality of configurations storedat the receiver, wherein each of the plurality of configurations “u”,identifies a set of P_(u) base sinusoid vectors on constant frequenciesand wherein P_(u) is the order of the configuration and is equal to thenumber of complex numbers of the compressed CSI if configuration u isselected, calculating, for each of the plurality of configurations, adot product of the N by 1 CSI vector and a conjugate of each P_(u) basesinusoid vector identified by the selected configuration to generate aP_(u) by 1 projection vector, calculating, for each of the plurality ofconfigurations, a product of a constant P_(u) by P_(u) matrix stored atthe receiver and the P_(u) by 1 projection vector to generate a P_(u) by1 coefficient vector and calculating, for each of the plurality ofconfigurations, a minimum squared error (MSE) fit with the P_(u) by 1coefficient vector on L evenly-spaced locations where L is smaller thanN, by multiplying each of the P_(u) base sinusoids with thecorresponding coefficient in the coefficient vector and taking thesummation, at each of the L evenly-spaced locations. The final step ofthe compression is to select configuration u and use its P_(u) by 1coefficient vector as the output, if the total fit residual of the MSEfit of configuration u is below a predetermined threshold times theminimum fit residual among all configurations, and u is such aconfiguration with the lowest order. The compressed CSI is transmittedto the transmitter of the wireless communication system, which maydecompress the CSI by computing a linear combination of the basesinusoids, based upon the decompressed CSI.

In another embodiment, the present invention provides a wirelesscommunication system for compressing channel state information (CSI) ofa wireless system. The system of the present invention includes, areceiver for receiving an orthogonal frequency division multiplexing(OFDM) wireless signal over a wireless channel. The receiver isconfigured for measuring a channel state information (CSI) vector of thewireless channel from the received OFDM wireless signal, wherein the CSIvector for each antenna pair is an N by 1 vector of complex numbers andwherein each complex number represents an amplitude and a phase of oneof N OFDM subcarriers of the wireless channel, accessing a plurality ofconfigurations stored at the receiver, wherein each of the plurality ofconfigurations “u” identifies a set of P_(u) base sinusoid vectors onconstant frequencies and wherein P_(u) is the order of the configurationand is equal to the number of complex numbers of the compressed CSI ifconfiguration u is selected. The receiver is further configured forcalculating, for each of the plurality of configurations, a dot productof the N by 1 CSI vector and a conjugate of each P_(u) base sinusoidvector identified by the selected configuration to generate a P_(u) by 1projection vector, for calculating, for each of the plurality ofconfigurations, for each of the plurality of configurations, a productof a constant P_(u) by P_(u) matrix stored at the receiver and the P_(u)by 1 projection vector to generate a P_(u) by 1 coefficient vector, forcalculating, for each of the plurality of configurations, a minimumsquared error (MSE) fit with the P_(u) by 1 coefficient vector on Levenly-spaced locations where L is smaller than N, by multiplying eachof the P_(u) base sinusoids with the corresponding coefficient in thecoefficient vector and taking the summation, at each of the Levenly-spaced locations. The receiver is further configured forselecting configuration u and use its P_(u) by 1 coefficient vector asthe output, if the total fit residual of the MSE fit of configuration uis below a predetermined threshold times the minimum fit residual amongall configurations, and u is such a configuration with the lowest order.The compressed CSI is transmitted to the transmitter of the wirelesscommunication system, which may decompress the CSI by computing a linearcombination of the base sinusoids, based upon the decompressed CSI.

In another embodiment, the present invention provides a non-transitorycomputer-readable recording medium storing a computer program used forexecuting a channel state information (CSI) compressing operation of anOrthogonal Frequency Division Multiplexing (OFDM) wireless channel of awireless communication system. The computer program causes a wirelesscommunication system to receive, at a receiver of the wirelesscommunication system, an OFDM wireless signal over a wireless channel,to measure, at a receiver of the wireless communication system, achannel state information (CSI) vector of the wireless channel from thereceived OFDM wireless signal for each antenna pair, wherein the CSIvector is an N by 1 vector of complex numbers and wherein each complexnumber represents an amplitude and a phase of one of N OFDM subcarriersof the wireless channel and to access a plurality of configurationsstored at the receiver, wherein each of the plurality of configurations“u” identifies a set of P_(u) base sinusoid vectors on constantfrequencies and wherein P_(u) is the order of the configuration and isequal to the number of complex numbers of the compressed CSI ifconfiguration u is selected. The computer program further causes thereceiver of the wireless communication system to calculate, for each ofthe plurality of configurations, a dot product of the N by 1 CSI vectorand a conjugate of each P_(u) base sinusoid vector identified by theselected configuration to generate a P_(u) by 1 projection vector, tocalculate, for each of the plurality of configurations, a product of aconstant P_(u) by P_(u) matrix stored at the receiver and the P_(u) by 1projection vector to generate a P_(u) by 1 coefficient vector, and tocalculate, for each of the plurality of configurations, a minimumsquared error (MSE) fit with the P_(u) by 1 coefficient vector on Levenly-spaced locations where L is smaller than N, by multiplying eachof the P_(u) base sinusoids with the corresponding coefficient in thecoefficient vector and taking the summation, at each of the Levenly-spaced locations. The computer program further causes thereceiver of the wireless communication system to select configuration uand use its P_(u) by 1 coefficient vector as the output, if the totalfit residual of the MSE fit of configuration u is below a predeterminedthreshold times the minimum fit residual among all configurations, and uis such a configuration with the lowest order. The computer programfurther causes the receiver to transmit the compressed CSI to thetransmitter of the wireless communication system. The computer programfurther causes the transmitter of the wireless communication system todecompress the CSI by computing a linear combination of the basesinusoids, based upon the compressed CSI.

The system and method of the present invention provides a novel CSIcompression method for wireless networks based upon a rigid mathematicalfoundation which provides, a high compression ratio with a low loss ofaccuracy, a very low computation complexity which is suitable forhardware implementation, a small range of compressed data that is easyto quantize and to transmit and a solution that is resistant to noise inthe channel.

The system and method employing the CSIApx technology in accordance withthe present invention serves as a strong candidate for a CSI compressionmethodology for future wireless protocols to be implemented in hardware,and significantly improve the capability of the wireless transmitter toacquire the most recent CSI and achieve higher transmission speed. Inaccordance with the present invention, a novel system and method tocompress the CSI vector in a wireless transceiver is described andreferred to herein as CSIApx.

Accordingly, the present invention provides an improved system andmethod for compressing the CSI for OFDM wireless signals that isaccurate and computationally easy to implement.

BRIEF DESCRIPTION OF THE DRAWINGS

For a fuller understanding of the invention, reference should be made tothe following detailed description, taken in connection with theaccompanying drawings, in which:

FIG. 1A is a graphical illustration of the bound of the coefficientswith an order of 5 of Direct Fit, in accordance with an embodiment ofthe present invention.

FIG. 1B is a graphical illustration of the bound of the coefficientswith an order of 7 of Direct Fit, in accordance with an embodiment ofthe present invention.

FIG. 1C is a graphical illustration of the bound of the coefficientswith an order of 11 of Direct Fit, in accordance with an embodiment ofthe present invention.

FIG. 1D is a graphical illustration of the bound of the coefficientswith an order of 16 of Direct Fit, in accordance with an embodiment ofthe present invention.

FIG. 2A is a graphical illustration of the fit residual of the DirectFit for a single target sinusoid with unit amplitude matched with anorder of 4, in accordance with an embodiment of the present invention.

FIG. 2B is a graphical illustration of the fit residual of the DirectFit for a single target sinusoid with unit amplitude matched with anorder of 6, in accordance with an embodiment of the present invention.

FIG. 2C is a graphical illustration of the fit residual of the DirectFit for a single target sinusoid with unit amplitude matched with anorder of 10, in accordance with an embodiment of the present invention.

FIG. 2D is a graphical illustration of the fit residual of the DirectFit for a single target sinusoid with unit amplitude matched with anorder of 16, in accordance with an embodiment of the present invention.

FIG. 3 illustrates locations of receivers and senders in variousexperimental locations, in accordance with embodiments of the presentinvention.

FIG. 4 is a graphical illustration of the absolute values of some rawCSI vectors, in accordance with embodiments of the present invention.

FIG. 5 is a graphical illustration of a typical fit by CSIApx inaccordance with embodiments of the present invention, where it can beseen that the fitted curve follows closely to the actual CSI.

FIG. 6A is a graphical illustration of the Cumulative Density Function(CDF) of the total fit residual of all 4 antenna pairs in 7923 CSImeasurements, in accordance with an embodiment of the present invention.

FIG. 6B is a graphical illustration of the compression ratio, defined asthe ratio of the number of real numbers in the CSI vector over thatneeded by a compression method to describe the sinusoids, noting that acomplex number consists of two real numbers, in accordance with anembodiment of the present invention.

FIG. 7 is a graphical illustration of the CDF of the normalized ratedifference in accordance with an embodiment of the present invention,where it can be seen that the rate difference with CSIApx is usuallyvery small.

FIG. 8 illustrates the distribution of the real and imaginary parts ofthe coefficients found by CSIApx for the strongest antenna pair in eachtest case, in accordance with an embodiment of the present invention.

FIG. 9A is a graphical illustration of a typical fit by CSIApx in atypical indoor environment with around 100 ns delay spread, inaccordance with an embodiment of the present invention.

FIG. 9B is a graphical illustration of a typical fit by CSIApx in atypical indoor environment with around 200 ns delay spread, inaccordance with an embodiment of the present invention.

FIG. 9C is a graphical illustration of a typical fit by CSIApx in atypical indoor environment with around 400 ns delay spread, inaccordance with an embodiment of the present invention.

FIG. 9D is a graphical illustration of a typical fit by CSIApx in atypical indoor environment with around 800 ns delay spread, inaccordance with an embodiment of the present invention.

FIG. 10A is a graphical illustration of the mean of the total fitresidual of all antenna pairs in a typical indoor environment witharound 100 ns delay spread, in accordance with an embodiment of thepresent invention.

FIG. 10B is a graphical illustration of the mean of the total fitresidual of all antenna pairs in a typical indoor environment witharound 200 ns delay spread, in accordance with an embodiment of thepresent invention.

FIG. 10C is a graphical illustration of the mean of the total fitresidual of all antenna pairs in a typical indoor environment witharound 400 ns delay spread, in accordance with an embodiment of thepresent invention.

FIG. 10D is a graphical illustration of the mean of the total fitresidual of all antenna pairs in a typical indoor environment witharound 800 ns delay spread, in accordance with an embodiment of thepresent invention.

FIG. 11A is a graphical illustration of the average compression ratiosin a typical indoor environment with around 100 ns delay spread, inaccordance with an embodiment of the present invention.

FIG. 11B is a graphical illustration of the average compression ratiosin a typical indoor environment with around 200 ns delay spread, inaccordance with an embodiment of the present invention.

FIG. 11C is a graphical illustration of the average compression ratiosin a typical indoor environment with around 400 ns delay spread, inaccordance with an embodiment of the present invention.

FIG. 11D is a graphical illustration of the average compression ratiosin a typical indoor environment with around 800 ns delay spread, inaccordance with an embodiment of the present invention.

FIG. 12A is a graphical illustration of the percentage of cases that thenormalized rate differences are above 3% or lower than −3% for anMU-MIMO rate in a typical indoor environment with around 100 ns delayspread, in accordance with an embodiment of the present invention.

FIG. 12B is a graphical illustration of the percentage of cases that thenormalized rate differences are above 3% or lower than −3% for anMU-MIMO rate in a typical indoor environment with around 200 ns delayspread, in accordance with an embodiment of the present invention.

FIG. 12C is a graphical illustration of the percentage of cases that thenormalized rate differences are above 3% or lower than −3% for anMU-MIMO rate in a typical indoor environment with around 400 ns delayspread, in accordance with an embodiment of the present invention.

FIG. 12D is a graphical illustration of the percentage of cases that thenormalized rate differences are above 3% or lower than −3% for anMU-MIMO rate in a typical indoor environment with around 800 ns delayspread, in accordance with an embodiment of the present invention.

FIG. 13A shows the distribution of the fit coefficients by CSIApx forthe strongest antenna pair in a typical indoor environment with around100 ns delay spread, in accordance with an embodiment of the presentinvention.

FIG. 13B shows the distribution of the fit coefficients by CSIApx forthe strongest antenna pair in a typical indoor environment with around200 ns delay spread, in accordance with an embodiment of the presentinvention.

FIG. 13C shows the distribution of the fit coefficients by CSIApx forthe strongest antenna pair in a typical indoor environment with around400 ns delay spread, in accordance with an embodiment of the presentinvention.

FIG. 13D shows the distribution of the fit coefficients by CSIApx forthe strongest antenna pair in a typical indoor environment with around800 ns delay spread, in accordance with an embodiment of the presentinvention.

FIG. 14 is a graphical illustration of the potential improvementutilizing Huffman Coding in addition to CSIApx, in accordance with anembodiment of the present invention.

FIG. 15A is a graphical illustration of the comparison between CSIApxand Givens rotation on the model data in a typical indoor environmentwith around 100 ns delay spread, in accordance with an embodiment of thepresent invention.

FIG. 15B is a graphical illustration of the comparison between CSIApxand Givens rotation on the model data in a typical indoor environmentwith around 200 ns delay spread, in accordance with an embodiment of thepresent invention.

FIG. 15C is a graphical illustration of the comparison between CSIApxand Givens rotation on the model data in a typical indoor environmentwith around 400 ns delay spread, in accordance with an embodiment of thepresent invention.

FIG. 15D is a graphical illustration of the comparison between CSIApxand Givens rotation on the model data in a typical indoor environmentwith around 800 ns delay spread, in accordance with an embodiment of thepresent invention.

FIG. 16 is a graphical illustration of the compression ratio for theexperimental data, in accordance with an embodiment of the presentinvention.

FIG. 17 is a graphical illustrating of a comparison of the data rateachieved in a MU-MIMO setting from both CSIApx and Givens Rotation,showing the percentage of cases where the normalized rate differences,in accordance with an embodiment of the present invention.

FIG. 18 is a block diagram illustrating a structure of a communicationsystem with channel state information (CSI) feedback, according to anembodiment of the present invention.

FIG. 19A is a diagram illustrating a first step of the functionality ofthe CSI compression circuitry of the receiver of the wirelesscommunication system, in accordance with an embodiment of the presentinvention.

FIG. 19B is a diagram illustrating a second step of the functionality ofthe CSI compression circuitry of the receiver of the wirelesscommunication system, in accordance with an embodiment of the presentinvention.

FIG. 19C is a diagram illustrating a third step of the functionality ofthe CSI compression circuitry of the receiver of the wirelesscommunication system, in accordance with an embodiment of the presentinvention.

FIG. 20. is a flow diagram illustrating a method for compression the CSIin a wireless communication system, in accordance with an embodiment ofthe present invention.

DETAILED DESCRIPTION OF THE INVENTION

In the following detailed description of the preferred embodiments,reference is made to the accompanying drawings, which form a parthereof, and within which are shown by way of illustration specificembodiments by which the invention may be practiced. It is to beunderstood that other embodiments may be utilized and structural changesmay be made without departing from the scope of the invention.

OFDM is the choice of modulation for broadband wireless networks today.In OFDM, entire communication frequency bandwidth is divided into equalsize chunks, where the center of each chunk is called a subcarrier. Asubcarrier can be considered as a pure sinusoid on a certain frequency,the amplitude and phase of which are set based on the values of the databits. The CSI of a subcarrier is basically the amplitude and phase ofthe subcarrier observed by the receiver, when the sender transmits theunchanged subcarrier. If there is only one path from the sender to thereceiver, the phases of the CSI of adjacent subcarriers will differ bythe same amount, which is the phase difference due to the difference infrequencies over the same path length. Therefore, the CSI of allsubcarriers will be a sinusoid. A typical wireless environment hasmultiple paths, therefore the CSI is the summation of multiplesinusoids.

Regarding the proof of the theorems supporting the present invention,the following detailed description proves that a target sinusoid onfrequency g can be approximated as the linear combination of P basesinusoids, within certain error, under certain conditions. Therefore,the summation of many sinusoids, such as the CSI vector, can still beapproximated as the linear combination of only P base sinusoids.

To approximate the target sinusoid, the existence of polynomialapproximation of sinusoids is used, such as those according to theTaylor series expansion, as a stepping stone. That is, if the targetsinusoid can be approximated by a polynomial of degree P−1, withincertain error, it can also be approximated by P base sinusoids, withincomparable error, because coefficients for the base sinusoids can befound by solving a linear system guaranteed to be nonsingular and thecoefficients are bounded by constants. Although it may appear from theproof that the sinusoidal approximation is only as good as thepolynomial approximation, in practice, it is preferred to use sinusoidsto approximate a sinusoid because they belong to the same functionfamily which significantly eases the numerical computations. Thefrequencies of the base sinusoids are not very sensitive to g, making itpossible to preselect constant base sinusoids to match a range of thetarget frequencies to simplify the computation. The linear combinationof the base sinusoids is called a fit, and the number of base sinusoidsthe order. The fit error refers to the difference between the fit andthe CSI vector, and the fit residual is the squared norm of the fiterror vector.

It can be said that, cos(Fx) with x in [0,1] can be approximated by apolynomial with degree P−1 and deviation ξ^(F,P), if a polynomial

${\Phi^{F,P}({Fx})} = {\sum\limits_{l = 0}^{P - 1}{\eta_{l}^{F,P}({Fx})}^{l}}$can be found such thatmax{|cos(Fx)−Φ^(F,P)(Fx)|}≦ξ^(F,P),where η_(l) ^(F,P) are constants determined by F and P. F isintentionally separated from η_(l) ^(F,P) because the same polynomialcan be reused for lower frequencies; that is, for any f<F, using Σ_(l=0)^(P−1)η_(l) ^(F,P)(fx)^(l) as the approximation for cos(fx) will lead todeviation no more than ξ^(F,P), because [0,f]⊂[0,F].

It is important to note that the proof does not need the exact values ofη_(l) ^(F,P), and only needs the existence of the polynomialapproximation. It is well established that polynomial approximationexists, such as the one according to the Taylor series expansion.Further optimizations can be made to minimize P for a given desireddeviation by solving minimum square error problems. However, polynomialapproximation can be difficult to find numerically in practice due tothe errors accumulated in the computation, because the optimizationproblem has to handle values with very large range, which is whysinusoidal approximation is preferred.

First focusing on the approximation of one target sinusoid.

Theorem 1

cos(gx) for xε[0,1] and g in [0,F] can be approximated with deviationξ^(F,P) as the linear combination of Φ^(F,P)(f₁x), Φ(f₂x), . . . ,Φ^(F,P)(f_(P)x), where f₁, f₂ . . . , f_(P) are distinct values in[0,F], and the coefficient of Φ^(F,P)(f_(k)x) in the linear combinationis

$\prod\limits_{{h = 1},{h \neq k}}^{P}{\frac{f_{h} - q}{f_{h} - f_{k}}.}$

Proof.

Clearly, if the linear combination of Φ^(F,P) (f₁x), Φ^(F,P) (f₂x), . .. , Φ^(F,P) (f_(P)x) can exactly reproduce Φ^(F,P) (gx), cos(gx) can beapproximated with deviation ξ^(F,P). Denote the coefficient forΦ^(F,P)(f_(k)x) as γ_(k) for 1≦k≦P. To reproduce Φ^(F,P)(gx) is to findcoefficients such that

${\sum\limits_{k = 1}^{P}{\gamma_{k}f_{k}^{l}}} = g^{l}$for all l; that is, the coefficients have to satisfy P linear equations:

$\begin{matrix}{{\begin{bmatrix}1 & 1 & \ldots & 1 \\f_{1} & f_{2} & \ldots & f_{P} \\f_{1}^{2} & f_{2}^{2} & \ldots & f_{P}^{2} \\\; & \vdots & \; & \; \\f_{1}^{P - 1} & f_{2}^{P - 1} & \ldots & f_{P}^{P - 1}\end{bmatrix}\begin{bmatrix}\gamma_{1} \\\gamma_{2} \\\gamma_{3} \\\vdots \\\gamma_{P}\end{bmatrix}} = \begin{bmatrix}1 \\g \\g^{2} \\\vdots \\g^{P - 1}\end{bmatrix}} & (1)\end{matrix}$

The matrix is a Vandermonde matrix with determinant

${\prod\limits_{1 \leq h < k \leq P}\left( {f_{k} - f_{h}} \right)};$therefore, as long as f₁, f₂, . . . , f_(P) are distinct, thedeterminate is not 0, and the solution always exists.

For convenience, denote f_(k)−g as d_(k). To obtain the values of γ_(k),it is first proved that for any γ_(k) that satisfies the linear systemin Eq. 1 must satisfy

$\begin{matrix}{{\begin{bmatrix}1 & 1 & \ldots & 1 \\d_{1} & d_{2} & \ldots & d_{P}^{1} \\d_{1}^{2} & d_{2}^{2} & \ldots & d_{P}^{2} \\\; & \vdots & \; & \; \\d_{1}^{P - 1} & d_{2}^{P - 1} & \ldots & d_{P}^{P - 1}\end{bmatrix}\begin{bmatrix}\gamma_{1} \\\gamma_{2} \\\gamma_{3} \\\vdots \\\gamma_{P}\end{bmatrix}} = \begin{bmatrix}1 \\0 \\0 \\\vdots \\0\end{bmatrix}} & (2)\end{matrix}$

To show this, note that the first equation in Eq. 2 is identical to thatin Eq. 1. For other equations, note that

$\begin{matrix}{{\sum\limits_{k = 1}^{P}{\gamma_{k}\left( {f_{k} - g} \right)}^{h}} = {\sum\limits_{k = 1}^{P}{\gamma_{k}{\sum\limits_{t = 0}^{h}\left\lbrack {\left( {- 1} \right)^{h - t}f_{k}^{t}g^{h - t}{C\left( {h,t} \right)}} \right\rbrack}}}} \\{= {\sum\limits_{t = 0}^{h}{\sum\limits_{k = 1}^{P}\left\lbrack {\left( {- 1} \right)^{h - t}\gamma_{k}f_{k}^{t}g^{h - t}{C\left( {h,t} \right)}} \right\rbrack}}} \\{= {\sum\limits_{t = 0}^{h}\left\lbrack {\left( {- 1} \right)^{h - t}g^{h - t}{C\left( {h,t} \right)}{\sum\limits_{k = 1}^{P}{\gamma_{k}f_{k}^{t}}}} \right\rbrack}} \\{= {\sum\limits_{t = 0}^{h}\left\lbrack {\left( {- 1} \right)^{h - t}g^{h - t}{C\left( {h,t} \right)}g^{t}} \right.}} \\{{= {\left( {g - g} \right)^{h} = 0}},}\end{matrix}$therefore the claim is proved. According to linear algebra, γ_(k) is theratio of the determinant of the following matrix

$M_{k} = \begin{bmatrix}1 & 1 & \ldots & 1 & 1 & 1 & \ldots & 1 \\d_{1} & d_{2} & \ldots & d_{k - 1} & 0 & d_{k + 1} & \ldots & d_{P} \\d_{1}^{2} & d_{2}^{2} & \ldots & d_{k - 1}^{2} & 0 & d_{k + 1}^{2} & \ldots & d_{P}^{2} \\\; & \vdots & \; & \; & \; & \; & \; & \; \\d_{1}^{P - 1} & d_{2}^{P - 1} & \ldots & d_{k - 1}^{P - 1} & 0 & d_{k + 1}^{P - 1} & \ldots & d_{P}^{P - 1}\end{bmatrix}$and the determinate of the matrix in Eq. 2. Using the cofactor expansionon column k, it can be found that

${\det\; M_{k}} = {\prod\limits_{{l \leq m \leq P},{m \neq k}}{d_{m}{\prod\limits_{{l \leq n < m \leq P},m,{n \neq k}}\left( {d_{m} - d_{n}} \right)}}}$

Therefore,

$\begin{matrix}{\gamma_{k} = {\overset{P}{\prod\limits_{{h = 1},{h \neq k}}}\frac{f_{h} - g}{f_{h} - f_{k}}}} & (3)\end{matrix}$

Theorem 1 basically shows that a target sinusoid can be approximated asthe linear combination of the polynomial approximations of the basesinusoids. However, in the present invention, the main interest is toapproximate the target sinusoid by the base sinusoids. An obviousapproach is to multiply base sinusoids by the coefficient values statedin Eq. 3, i.e., replacing the polynomial approximations by the basesinusoids themselves, and use the summation as the approximation, whichis referred to as the Direct Fit. The Direct Fit will also have boundederror, if the coefficients are bounded, which is proved by the followingtheorem.

Theorem 2

Assume the frequencies of the base sinusoids are evenly distributed,i.e.,

$f_{k} = {\frac{\left( {k - 1} \right)F}{P - 1}.}$Let

${g = \frac{\left( {h + \delta} \right)F}{P - 1}},$where δε[0,1] and h is an integer in [0,P−2].

For

${h < \frac{P - 1}{2}},$|γ_(k)| is bounded by Θ(g,k) where

${\Theta\left( {g,k} \right)} = \left\{ \begin{matrix}\frac{{\left\lbrack {\left( {h + 1} \right)!} \right\rbrack\left\lbrack {\left( {P - 1 - h} \right)!} \right\rbrack}^{\frac{P - h}{P - 1 - h}}\left( {P - 1 - h} \right)^{P - 1 - h}}{\left( {h - k + 2} \right){\left( {k - 1} \right)!}{\left( {P - k} \right)!}\left( {P - h} \right)^{P - h}} \\1 \\\frac{{\left\lbrack {\left( {h + 1} \right)!} \right\rbrack\left\lbrack {\left( {P - 1 - h} \right)!}^{\frac{P - 1 - h}{P - 2 - h}} \right\rbrack}\left( {P - 2 - h} \right)^{P - 2 - h}}{{\left( {k - 1} \right)!}{\left( {P - k} \right)!}\left( {P - 1 - h} \right)^{P - 1 - h}\left( {k - 1 - h} \right)^{\frac{P - 1 - h}{P - 2 - h}}}\end{matrix} \right.$for k<h+1, k=h+1, and k>h+1, respectively. For

${h \geq \frac{P - 1}{2}},$due to symmetry, Θ(g,k)=Θ(F−g,P−k+1).

Proof.

It is first noted that Eq. 3 cancels all common terms between thenumerator and the denominator, therefore, it is convenient to normalizef_(k) to k−1 and g to h+δ in this proof. If k≦h+1,

${\gamma_{k}} = {\frac{\left\lbrack {\prod\limits_{{t = 1},{t \neq {h - k + 1}}}^{h}\left( {t + \delta} \right)} \right\rbrack{\delta\left\lbrack {\prod\limits_{t = 1}^{P - 1 - h}\left( {t - \delta} \right)} \right\rbrack}}{{\left( {k - 1} \right)!}{\left( {P - k} \right)!}} = \frac{\left\lbrack {\prod\limits_{{t = 1},{t \neq {h - k + 1}}}^{h}\left( {t + \delta} \right)} \right\rbrack\left\lbrack {\prod\limits_{t = 1}^{P - 1 - h}{\delta^{\frac{1}{P - 1 - h}}\left( {t - \delta} \right)}} \right\rbrack}{{\left( {k - 1} \right)!}{\left( {P - k} \right)!}}}$

As

$\delta^{\frac{1}{P - 1 - h}}\left( {t - \delta} \right)$is maximized when

${\delta = \frac{t}{P - h}},$results in

${\gamma_{k}} \leq \frac{{\left\lbrack {\left( {h + 1} \right)!} \right\rbrack\left\lbrack {\left( {P - 1 - h} \right)!} \right\rbrack}^{\frac{P - h}{P - 1 - h}}\left( {P - 1 - h} \right)^{P - 1 - h}}{\left( {h - k + 2} \right){\left( {k - 1} \right)!}{\left( {P - k} \right)!}\left( {P - h} \right)^{P - h}}$If k=h+1,

${\gamma_{h + 1}} = {\frac{\left( {h + \delta} \right){\ldots\left( {1 + \delta} \right)}\left( {1 - \delta} \right){\ldots\left( {P - 1 - h - \delta} \right)}}{{h!}{\left( {P - 1 - h} \right)!}} = {\frac{\left\lbrack {\prod\limits_{t = 1}^{h}\left( {t^{2} - \delta^{2}} \right)} \right\rbrack\left\lbrack {\prod\limits_{t = {h + 1}}^{P - 1 - h}\left( {t - \delta} \right)} \right\rbrack}{{h!}{\left( {P - 1 - h} \right)!}} \leq 1}}$where the maximization occurs when δ=0. If k>h+1,

${\gamma_{k}} = {\frac{\left\lbrack {\prod\limits_{t = 1}^{h}\left( {t + \delta} \right)} \right\rbrack{\delta\left\lbrack {\prod\limits_{{t = 1},{t \neq {k - 1 - h}}}^{P - 1 - h}\left( {t - \delta} \right)} \right\rbrack}}{{\left( {k - 1} \right)!}{\left( {P - k} \right)!}} = \frac{\left\lbrack {\prod\limits_{t = 1}^{h}\left( {t + \delta} \right)} \right\rbrack\left\lbrack {\prod\limits_{{t = 1},{t \neq {k - 1 - h}}}^{P - 1 - h}{\delta^{\frac{1}{P - 2 - h}}\left( {t - \delta} \right)}} \right\rbrack}{{\left( {k - 1} \right)!}{\left( {P - k} \right)!}}}$

Using the same bounding technique as for k<h+1, it can be shown that

${\gamma_{k}} \leq \frac{{\left\lbrack {h + {1!}} \right\rbrack\left\lbrack {\left( {P - 1 - h} \right)!}^{\frac{P - 1 - h}{P - 2 - h}} \right\rbrack}\left( {P - 2 - h} \right)^{P - 2 - h}}{{\left( {k - 1} \right)!}{\left( {P - k} \right)!}\left( {P - 1 - h} \right)^{P - 1 - h}\left( {k - 1 - h} \right)^{\frac{P - 1 - h}{P - 2 - h}}}$

Theorem 2 bounds the amplitudes for one particular choice the basesinusoids, which is sufficient to prove the existence of boundedapproximation. The bounding technique in the proof is simple, however isfairly good and usually is within a small factor of 2 or 3 to the actualvalue. It can be easily found that the coefficient absolute values areno more than 1 when P is 3 or less; for other typical values, FIG. 1shows the bound according to Theorem 2, where it can be seen that thevalues are usually small, except for target frequencies near the ends,which still does not pose a serious problem in practice according to theevaluation, likely because most frequency components in these regionsare very weak.

Combining Theorems 1 and 2, if cos(gx) is approximated by the Direct Fitwith evenly spaced base sinusoids, the total deviation is bounded by thedifference between cos(gx) and Φ^(F,P) (gx), which is bounded byξ^(F,P), plus the deviation of the polynomial approximation of thescaled bases, which for base k is bounded by ξ^(F,P)Φ(g,k). Therefore:

Theorem 3

There exists a sinusoidal approximation for cos(gx) with deviationξ^(F,P)[1+Σ_(k=1) ^(P)Θ(g,k)] using P base sinusoids, where gε[0,F],xε[0,1], and the base sinusoids have evenly spaced frequencies in [0,F].

It is important to note that as FIG. 1A-FIG. 1C suggests, the deviationaccording to Theorem 3 can be fairly small for a wide range of targetfrequencies, even when base sinusoids are on fixed and evenly spacedfrequencies. In other words, it is not needed in practice to adjust thefrequencies of the base sinusoids to match a target sinusoid to maintaina small approximation error, which can significantly simplify theoperations in practice because many steps, such as finding the inverseof a matrix, can be pre-computed.

Several extensions to the above theorems will now be discussed. Thefocus so far has been on cos(gx) for x in [0,1]. However, the extensionto [−1,1] is immediate. That is, if the approximation matches cos(gx) in[0,1], multiplying the base sinusoids by the same coefficients forvalues in [−1,0] should also result in a match, because the targetsinusoid and the base sinusoids have the same parity. Although thisextension is very simple, it in effect doubles the range of thefrequencies that can be approximated by P sinusoids.

The approximation for sin(gx) can use exactly the same coefficients asthose for cos(gx). That is, given an approximation of cos(gx)≈Σ_(k=1)^(P)γ_(k) cos(f_(k)x) for xε[−1,1], sin(gx)≈Σ_(k=1) ^(P)γ_(k)sin(f_(k)x) for xε[−1,1]. This is because although the cosine and sinefunctions have different polynomial approximations, as long as Eq. 1 issatisfied, the coefficients for any exponent of x in the polynomialapproximation are the same for target sinusoid and the linearcombination of the base sinusoids.

The extension to complex wave e^(igx) for xε[−1,1] is

${{\sum\limits_{k = 1}^{P}{\gamma_{k}{\cos\left( {f_{k}x} \right)}}} + {i\;{\sum\limits_{k = 1}^{P}{\gamma_{k}{\sin\left( {f_{k}x} \right)}}}}} = {\sum\limits_{k = 1}^{P}{\gamma_{k}e^{{if}_{k}x}}}$

The CSI vector is the summation of more than one sinusoid. However, aseach sinusoid can be approximated, the summation can also beapproximated as the summation of the individual approximations. Thedeterministic maximum deviation will be the summation of all individualdeviations multiplied by the amplitude of the individual sinusoids, andmay be large. However, in practice, the sinusoids have different phasesand the deviation will almost never add up constructively, therefore theapproximation error is small.

FIG. 2A-FIG. 2D shows the fit residual of the Direct Fit for a singletarget sinusoid with unit amplitude in several frequency ranges matchedwith different orders. The frequency is the amount of angular rotationbetween neighboring points and the target sinusoid vector has a total of64 points with index from −32 to 31, matching the number of subcarriersin Wi-Fi. The frequencies of the base sinusoids are evenly spacedbetween 0 and the maximum frequency in each case. The orders in thefigure are larger than the orders actually used in CSIApx for the samefrequency range, because the Direct Fit is not optimal. Still, it can beseen that the fit residual is very small for most target sinusoidfrequencies, such as around 0.01 or lower, which translates to only0.00016 per data point. The residual is higher for target frequencies inthe ends, matching our theoretical prediction. The good fit quality alsoindirectly confirms the existence of polynomial approximation with orderP−1, because the Direct Fit matches the target sinusoid exactly inorders P−1 and lower in the polynomial approximation, therefore theerror is contributed by the higher orders; as the error is small, thecoefficients in the higher orders are bounded.

According to our theoretical findings, the CSI can be approximated by alinear combination of the base sinusoids. In practice, the coefficientsof the base sinusoids can be found by solving a minimum square errorproblem to match the CSI as closely as possible. The fit is thereforecalled the MSE Fit, and requires very low computation complexity, mainlybecause the linear system to be solved in the optimization problem isdefined by a constant matrix. As any sinusoid can be approximated inthis manner, the simplest approach would be to select just one set ofbase sinusoids to be used for all CSI. However, different types ofchannels have different delay spreads, which translate to differentfrequency ranges of the sinusoids in the CSI, the larger the delayspread, the higher the frequency. As sinusoids on lower frequencies canbe approximated with fewer base sinusoids, to further improve thecompression ratio, CSIApx introduce a small number of configurationswith different number of base sinusoids, and finds the fits for allconfigurations and selects a fit as the output, considering bothcompression ratio and fit residual. The complexity of calculating thefits are reduced by sharing certain base sinusoids among multipleconfigurations.

The very core of CSIApx is to find the coefficients of the MSE Fit,denoted as a vector Γ. Let Y denote the CSI vector and N the length ofY. The following explanation is for the case where the CSI vector ismeasured at N consecutive subcarriers. However, the same method can beeasily extended to the case where the subcarriers are not consecutive byminimizing squared error only at the subcarriers with measurements; thatis, fitting only the measured subcarriers. To minimize the squared erroris to select coefficients to minimizeJ=Σ _(j=1) ^(N)|(Σ_(k=1) ^(P)γ_(k) e ^(ijf) ^(k) )−γ_(j)|²,where P is the order, γ_(k) and f_(k) are the coefficient and frequencyof base sinusoid k, respectively, y_(j) is element j in the CSI vectorY. By taking the derivatives of J with respect to the coefficients andsetting them to 0, Γ that minimizes J is the solution to a linear systemQΓ=S, where:

-   -   Q is a P by P matrix, in which q_(k,h)=Σ_(j=1) ^(N)e^(i(f) ^(h)        ^(−f) ^(k) ^()j),    -   S is a P by 1 vector, in which s_(k)=Σ_(j=1) ^(N)e^(−if) ^(k)        ^(j)y_(j).

It can be seen that S is determined by the CSI vector, but is just thedot products of the CSI vector and the conjugate of the base sinusoids.On the other hand, as the frequency values are constants, Q⁻¹ is aconstant matrix and can be precomputed. Therefore, after S is obtained,Γ can be found by simply multiplying the constant matrix Q⁻¹ with S.

In an additional embodiment, CSIApx introduces U configurations to matchWi-Fi channel conditions. Configuration u is described by thefrequencies of the P_(u) base sinusoids it uses, denoted as f_(u,k) fork=1, 2, . . . , P_(u), where the definition of frequency is aspreviously defined.

CSIApx solves the constant linear system previously described to get thefit coefficients for each configuration, in parallel. Then, itcalculates the MSE Fit on L evenly spaced sample locations for eachconfiguration and finds the sampled fit residual for each configuration,denoted as η_(u) for configuration u. CSIApx selects the fitcoefficients of configuration u as the compressed CSI, if configurationu is the first configuration such that η_(u)<ζmin(η₁, η₂, . . . ,η_(U)), where (is an empirical constant greater than 1.

To reduce the overall computation complexity, different configurationsmay share some common base sinusoids, because the main computation inCSIApx is actually finding the dot products between the base sinusoidsand the CSI. Overall, let W be the total number of unique base sinusoidsin all configurations, the complexity of CSIApx, measured the number ofcomplex multiplications, includes only:

-   -   WN: for computing the dot products between the base sinusoids        and the CSI    -   Σ_(u=1) ^(U) P_(u) ²: for finding the fit coefficients of all        configurations    -   Σ_(u=1) ^(U) (P_(u)+1)L: for computing the fits at sampled        locations and the sampled fit residuals

The selection of the base sinusoids should consider multiple issues thatare dependent on each other, such as compression ratio, accuracy,implementation cost, and the range of the fit coefficients. Fortunately,CSIApx is to be applied to wireless systems with a fixed number ofsubcarriers and well-studied channel conditions, such as Wi-Fi.Therefore, the parameters can be empirically chosen. As such, based onexperience, for configuration u, the maximum base sinusoid frequencyshould be the maximum frequency P_(u) base sinusoids can approximatewell. Also, for each configuration, the base sinusoids should be moreclosely spaced in the lower part of the spectrum than the upper part,because more energy is in the lower part of the spectrum.

Table 1 shows exemplary choices targeting Wi-Fi channel with 64subcarriers, in which there are 5 configurations with 27 unique basesinusoids, and ζ=1.75. For 40 subcarriers used for evaluating theexperimental data, the frequencies for configurations 1 to 5 are shownin Table 2, there are 5 configurations with 27 unique base sinusoids andζ=4.

TABLE 1 Configuration For 64 subcarriers u P_(u) Frequencies 1 3 0,0.06, 0.12 2 5 0, 0.05, 0.1, 0.15, 0.25 3 7 0, 0.06, 0.12, 0.18, 0.24,0.3, 0.42 4 11 0, 0.06, 0.12, 0.18. 0.24, 0.3, 0.36, 0.42, 0.525,0.6375, 0.75 5 16 0, 0.075, 0.15, 0.225, 0.3, 0.375, 0.45, 0.525, 0.6,0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3

TABLE 2 Configuration For 40 subearriers u P_(u) Frequencies 1 3 0,0.05, 0.1 2 4 0, 0.06, 0.12, 0.2 3 6 0, 0.075, 0.15, 0.225, 0.3, 0.45 410 0, 0.075, 0.15, 0.225, 0.3, 0.375, 0.525, 0.675, 0.825, 0.975 5 14 0,0.09, 0.18, 0.27, 0.36, 0.45, 0.575, 0.7, 0.825, 0.95, 1.075, 1.2,1.325, 1.45

In practice, the raw measured CSI often has a shift frequency, which isa frequency value added to the frequencies of all sinusoids, caused bythe sample timing offset to the OFDM symbol boundary. The shiftfrequency needs to be removed before running CSIApx, because it mayforce CSIApx to choose higher configurations to approximate sinusoids onhigher frequencies and reduce the compression ratio. This can easily beachieved by multiplying the CSI with a sinusoid on the negative of theshift frequency, a process referred in the invention as “rotation”. Thevalue of the shift frequency is known to the wireless receiver, becauseit selects the OFDM symbol boundary. The frequency used in the rotationcan also be slightly adjusted to make sure that the sinusoids in the CSIare still on positive frequencies after the rotation.

The implementation guideline of CSIApx described herein summarizes theearlier discussions and is for a given number of subcarriers in theWi-Fi channel, denoted by N. The following explanation is for the casewhere the CSI vector is measured at N consecutive subcarriers; however,it can be easily extended to the case where the subcarriers are notconsecutive, as explained earlier. It should also be noted that the samemethod applies to all possible values of N, except the values of theconstants need to be adjusted.

The CSI vector Y is an N by 1 vector of complex numbers, where eachnumber represents the amplitude and phase of the OFDM subcarrier. CSIApxsupports U configurations. Configuration u selects a set of P_(u) basesinusoids, which are sinusoids on constant frequencies, denoted asf_(u,k) for k=1, 2, . . . , P_(u). To reduce the complexity, differentconfigurations may share some common base sinusoids. If configuration uis selected by CSIApx, Y will be compressed into P_(u) complex numbers.P_(u) is called the order of configuration u.

The preprocessing steps of CSIApx are performed only once. A firstpreprocessing step includes, computing the original and conjugate of thebase sinusoid vectors for all configurations. For example, the originalbase sinusoid vector k for configuration u is a sinusoid on frequencyf_(u,k) evaluated on N points. For any base sinusoid shared by multipleconfigurations, such computation is to be performed only once. In asecond preprocessing step, for each configuration, say, configuration u,compute a P_(u) by P_(u) matrix Q_(u) ⁻¹ which is a constant matrix andis the inverse of matrix Q_(u), where element (k,h) is Σ_(j=1)^(N)e^(i(f) ^(u,h) ^(−f) ^(u,k) ^()j).

Various methods are known in the art for measuring the CSI of the OFDMwireless signal at the receiver. The procedure of actual CSI measurementat the receiver is beyond the scope of the present invention. Thepresent invention can successfully operate with any method of real-timeCSI measurement, including, but not limited to, pilot based CSImeasurement, blind SCI measure and data assisted n pilot based CSImeasurement.

After using the existing method to measure the CSI of the OFDM wirelesssignal at the receiver, a first real-time compression step includes, foreach configuration, say, configuration u: (a) Obtain the dot product ofY with the conjugate of each base sinusoid. Note that for any basesinusoid shared by multiple configurations, such computation is to bedone only once. Store the results in a P_(u) by 1 vector S_(u), (b)Compute Γ_(u)=Q_(u) ⁻¹S_(u). Element k in Γ_(u) is denoted as T_(u,k),and (c) Compute the sampled fit at L evenly spaced locations. Forexample, for a location h, it is Σ_(k=1) ^(P) ^(u) T_(u,k)e^(if) ^(u,k)^(h). Find the squared error of the sampled fit to Y at the sampledlocations denoted as θ_(u). The second real-time compression stepincludes, picking Γ_(u) as the compressed CSI, if configuration u is thefirst configuration such that η_(u)u<ζmin{η₁, η₂, . . . , η_(U)}, whereζ is an empirical constant greater than 1.

In various experimental embodiments, CSI data was collected using theAtheros CSITool installed on 2 laptops with the Atheros AR9462 wirelesscard with 2 antennas on 20 MHz channels. A total of 100 experiments invarious location settings were conducted, which include typicalenvironments like office buildings, apartment complexes, and largehallways. The experiments include both line of sight and non-line ofsight cases as well as varying channel conditions due to human movementsnear the machines. Some of the experiments locations are shown in FIG.3.

The CSITool reports the CSI on 56 selected subcarriers for 4 antennapairs. FIG. 4 shows the absolute values of some raw CSI vectors, whereit can be seen that the data has some level of noise. A fewpreprocessing steps were taken before passing the data for compression.First, as the signal always seems to be attenuated at both ends of thespectrum, caused most likely by additional filtering in hardware, notrepresenting the characteristics of the actual channel, 8 subcarriers onboth ends are removed, with only the middle 40 subcarriers kept.Secondly, as some of the experiments have very weak signals,measurements with RSSI (Received Signal Strength Indicator) lower than30 dB are filtered out. Thirdly, the CSI data of all antenna pairs isnormalized by a common factor such that the maximum amplitude is 1.Lastly, as explained before, each CSI vector is rotated to remove theshift frequency. A simple heuristic is used to estimate the shiftfrequency value, which is not reported by the current device to thedriver level. The details of the heuristics is omitted; basically, itkeeps rotating the CSI from the same transmitting antenna incrementally,until most energy appears to occupy only a spectrum starting from 0 upto some frequency for both receiving antennas. As it may over-rotate theCSI and lead to negative frequencies, when running CSIApx, the CSI forall antenna pairs are multiplied by a sinusoid with a positive frequencyof 0.0491 to move most sinusoids in the CSI to positive frequency.

For comparison, the known CTDP (Continuous Time Domain Parameters)channel estimation was implemented. The CTDP iteratively selects asinusoid that best matches the current residual signal, until the powerof the selected sinusoid is below a threshold. CTDP is chosen because itis one of the more recent methods and has a good performance. As CTDPrequires the noise power value, which needs to be estimated with theexperimental data and the original CTDP design does not specify how thenoise level is estimated, the fit residual found by CSIApx is used asthe estimate of the total noise power, which should be very close.Another constrained version of CTDP, referred to as cCTDP, is alsoevaluated, with which the fit residuals of CTDP and CSIApx can becompared when using similar number of sinusoids. That is, with cCTDP,the number of sinusoids used is the smallest upper bound of the averagenumber of sinusoids used by CSIApx for the same CSI measurement, notingthat CSIApx may use different configurations for different antennapairs. The frequency range of sinusoids scanned in CTDP and cCTDP is[−0.785,157], which should cover all frequencies in the CSI. It shouldalso be mentioned that as CTDP has to solve an optimization problem toselect the frequency of a sinusoid in each iteration, it has much higherimplementation complexity than CSIApx, because CSIApx avoids thisproblem altogether by using constant frequencies.

FIG. 5 shows a typical fit by CSIApx, where it can be seen that thefitted curve follows closely to the actual CSI. As the fit residual andthe compression ratio are related, i.e., improving one is often at thecost of the other, they are jointly compared.

FIG. 6A shows the Cumulative Density Function (CDF) of the total fitresidual of all 4 antenna pairs in 7923 CSI measurements. FIG. 6B showsthe compression ratio, defined as the ratio of the number of realnumbers in the CSI vector over that needed by a compression method todescribe the sinusoids, noting that a complex number consists of tworeal numbers.

It can be seen that the fit residual of CSIApx in most cases are verysmall with a median of 0.0828 for all antenna pairs, which translates toan error of 0.0005 per data point. The fit residual of CTDP is betterwith a median of 0.0467, however it is at a cost of a much lowercompression ratio, as the average compression ratio of CSIApx against 40subcarriers is 7.68, much better than CTDP, which is 3.59. Closeexaminations of the fits show that CSIApx actually fits the signal verywell, and its fit residual is mainly the quantization noise, such asthose shown in FIG. 6, which cannot be eliminated unless more sinusoidsare introduced to fit the noise. In this sense, CSIApx achieves a bettertradeoff between fit residual and compression. The better performancewith CSIApx can also be seen from the cCTDP results, as cCTDP actuallyhas higher fit residual, at the same time slightly lower compressionratio.

The end result of the compression method can be the MU-MIMO data rate ofthe users. In one embodiment, the MU-MIMO rate program is used, whichfirst calculates the modulation parameters with the supplied imperfectCSI, then finds the achievable data rate when the selected parametersare used on the actual channel. The program is configured to use thegreedy method for user selection and run at SNR of 20 dB. For eachsubcarrier, the program is run twice, feeding the compressed and themeasured CSI to the program to obtain two values, representing totaldata rates to all users with imperfect and perfect CSI, respectively.The difference between the two, divided by the latter, is referred to asthe “normalized rate difference”, and is used as the metric.

In the experimental tests, a total of 1500 tests were run, where eachtest includes one sender and 2 receivers. In each test, the CSIcollected from experiments where the sender was at a fixed location for4 receivers is used and 2 receivers are randomly selected from the 4actual receivers. As the link is 2 by 2 but each MU-MIMO receiver hasonly 1 antenna, the first antenna is selected for each receiver.

FIG. 7 illustrates the CDF of the normalized rate difference, where itcan be seen that the rate difference with CSIApx is usually very small,e.g., within −3% and 3% in over 98.3% of the cases. CTDP performs betterreporting 99.0%, but this comes at the cost of it's compression ratio.At similar compression ratio, cCTDP performs worse than CSIApx at 95.7%.The rate difference in some very rare cases can also be positive,because the greedy method sometimes selects different sets of users whengiven the compressed and measured CSI.

One of the advantageous features of CSIApx is that the fit coefficientsstay in a small range, making it simple and inexpensive to quantize andtransmit the coefficients as the compressed CSI. FIG. 8 shows thedistribution of the real and imaginary parts of the coefficients foundby CSIApx for the strongest antenna pair in each test case, because thedistributions for other antenna pairs should just be its scaledversions. It can be seen that all numbers reside in a small range withsmooth density.

In an additional exemplary embodiment, CSIApx is further tested withsynthesized CSI data, which complements the experimental evaluation bychallenging CSIApx with more channel types and testing CSIApx undercontrollable settings such as the Signal to Noise Ratio (SNR) level.

In the exemplary embodiment, the known channel model code is used togenerate CSI for all 64 subcarriers in Wi-Fi on 20 MHz channels for 3 by3 links with 9 antenna pairs. Four cases, referred to as Model B, ModelC, Model D and Model E, are used, which represent typical indoorenvironments with around 100 ns, 200 ns, 400 ns, and 800 ns delayspread, respectively, which should cover the majority of typical Wi-Fichannels.

As in the experimental data, the maximum amplitude is normalized to 1.White Gaussian noise was added to the CSI vector and a total of 1000test cases were performed for each SNR level. To simulate imperfectrotation, the CSI is further multiplied by a sinusoid with frequencyrandomly selected form [−0.0491,0.0491], which translate to within ±25ns of timing error. The frequency range of sinusoids scanned in CTDP andcCTDP is [−0.0491,1.57], which includes all frequencies in the CSI. Aswith the experimental data, when evaluating CSIApx, the CSI for allantenna pairs are multiplied by a sinusoid with a positive frequency of0.0491.

FIG. 9A-FIG. 9D shows a typical fit by CSIApx for each model, where itcan be seen that the fitted curve follows very closely to the CSI. Asthe clean CSI is available, when calculating the final fit residual, thefit is compared with the clean CSI; all prior steps are still based onthe noisy CSI. FIG. 10A-FIG. 10D shows the mean of the total fitresidual of all antenna pairs in various settings. The fit residual ofCSIApx is usually very small, such as about 0.0007 or lower per point at20 dB or above. In addition, as the noise level reduces by 5 dB, the fitresidual in most cases also reduces by roughly 5 dB, suggesting that thefit residual is mostly noise. FIG. 11A-FIG. 11D shows the averagecompression ratios. It can be seen that CSIApx achieves very highcompression ratios in many cases, i.e., above 12.4:1, 7.9:1, 5.5:1, and4.0:1 against 64 subcarriers for Models B, C, D, E, respectively, whenthe SNR is 20 dB or above. More complicated channel conditions do pose achallenge to CSIApx as it has to use higher orders to fit the data.Also, although CSIApx may have slightly larger fit residual, it has muchhigher compression ratios than CTDP in all cases. In addition, thecompression ratio CSIApx is more stable than CTDP for each model whenthe SNR is 20 dB or higher, suggesting the CSIApx is better at capturingthe actual signal and less susceptible to the influence of noise. cCTDPhas higher fit residual and lower compression ratios in almost all caseswhen the SNR is 20 dB or higher.

MU-MIMO rate was also tested in a similar manner as with theexperimental data. FIG. 12A-FIG. 12D shows the percentage of cases thatthe normalized rate differences are above 3% or lower than −3%, where itcan be seen that the fraction is very low for CSIApx when the SNR is 25dB or higher, and still reasonably small at 20 dB except for Model Ewhich is the most complicated.

FIG. 13A-FIG. 13D shows the distribution of the fit coefficients byCSIApx for the strongest antenna pair, which is similar to that with theexperimental data.

Even higher compression ratio can be achieved for CSIApx by runningHuffman coding on the coefficients, taking advantage of the fact thatthe distribution of the real and imaginary parts of the coefficients,such as that in FIG. 8, is spiky and has low entropy. The process isexplained for the experimental data in the following. First, 12 bits forquantization in range [−2.56,2.56] are empirically chosen, which resultsin negligible quantization error and includes all coefficients. Atraining set with 3962 experiments is randomly chosen from the data toobtain the dictionary of the Huffman Coding, which is then tested on theremaining data. FIG. 14 shows the CDF of the compression ratio achievedby Huffman Coding, where the ratio is calculated by subtracting the sizeof the raw coefficients by the size of the compressed coefficients thendivided by former. The average compression ratio is 22.1% and the ratiois positive for over 98% of the cases. Separate Huffman Codingdictionaries can also be built for each configuration; however, theresults show that the improvement is marginal.

The Wi-Fi standard includes an option to use the Givens Rotation tocompress CSI to be sent during the channel sounding procedure. That is,instead of sending the entire CSI, it calculates a compressed feedbackmatrix by zeroing out some elements, which is later reconstituted toobtain the full CSI. As detailed below, the present invention provides ahead-to-head comparison between CSIApx and the Given's rotation method,and argues that CSIApx is a better alternative.

Givens Rotation is lossless in the sense that it does not change the CSIduring the compression, and the other end of the communication link canexactly reproduce the measured CSI. It therefore appears that Given'srotation will have higher accuracy than CSIApx, because CSIApx is basedon approximation. This is true when the measured CSI is clean, i.e.,without any noise. However, measurement noise and quantization noisealways exist. In such cases, CSIApx actually achieves better accuracythan the Given's rotation, i.e., the CSI with CSIApx follows the shapeof the actual CSI more closely than the Givens Rotation, which iscorrupted by noise. To be more fair, before applying Given's Rotation,certain filter can be applied to reduce the noise level. Still, it isfound that CSIApx achieves better accuracy, which, from a high level, isbecause when fitting a CSI curve, CSIApx also serves as a filter, whichis better than other general purpose filters.

In the following comparison, the model data is used, because the cleanCSI is available for comparison. Before the Givens rotation, a low-passfilter is used in an attempt to filter out some noise, as it is expectedthat such filter will be used in practice. Due to the low pass filter,only the middle 50 subcarriers are used in this comparison. FIG.15A-FIG. 15D shows CSIApx and Givens Rotation on the model data, whereit can be seen that CSIApx indeed achieves lower fit residual. CSIApxwill also enjoy a higher compression ration than the Givens rotation.FIG. 16 shows the compression ratio for the experimental data. Asmentioned earlier the average compression ratio achieved by CSIApx forthe 2×2 system in a real world setting was 7.68. Because CSIApx fitseach antenna pair individually and hence can remain constant even whenthe number of pairs increases. The compression ratio achieved by Givensrotation on the other hand will keep decreasing as the antenna orderincrease approaching 2.

The first accuracy is further evaluated by comparing the data rateachieved in a MU-MIMO setting fro both CSIApx and Givens Rotation. FIG.17 shows the percentage of cases where the normalized rate differencesis higher than 3% or lower than −3% when compared against the cleansignal. It is seen that CSIApx outperforms Givens Rotation.

FIG. 18 is a block diagram illustrating a structure of a wirelesscommunication system 100 with channel state information (CSI) feedback,according to an embodiment of the present invention. In the illustratedwireless communication system 100, a transmitter 200 transmits awireless signal to a receiver 300 over a wireless channel, as iscommonly known in the art. The receiver 300 receives the wireless signaland the CSI measurement circuitry 310 at the receiver measures the CSIof the wireless channel, as previously described. The CSI compressioncircuitry 320 then compresses the CSI and transmits the compressed CSIcoefficient vector back to the transmitter 200. The transmitter 200 isthen able to adjust the transmission of the wireless signal based uponthe measured CSI of the wireless channel.

FIG. 19A is a diagram illustrating a first step in the functionality ofthe CSI compression circuitry 320 of the receiver 300. As shown, the CSIcompression circuitry 320 receives the measured N by 1 CSI vector 400from the CSI measurement circuitry 310, wherein the CSI measurementcircuitry 310 measures the channel state information (CSI) vector fromthe received OFDM wireless signal for each of one or more antenna pair.The CSI compression circuitry 320 then calculates for each of aplurality of configurations their projection vectors, wherein each ofthe plurality of configurations “u” identifies a set of P_(u) basesinusoid vectors on constant frequencies and wherein P_(u) is the orderof the configuration and is equal to the number of complex numbers ofthe compressed CSI if configuration u is selected. As the configurationsmay share common base sinusoids to reduce computation, the figure showsmultiplying the collection of the conjugate of all unique base sinusoidsin CSIApx 410 with the CSI 400 to get the collection of dot products forall unique base sinusoids 420.

In a second step illustrated with reference to FIG. 19B, the CSIcompression circuitry 320 then finds for each of the plurality ofconfigurations “u” P_(u) values in the collection of dot products 420 toform a P_(u) by 1 projection vector, and calculates the product of aconstant P_(u) by P_(u) matrix 425 stored a the receiver and the P_(u)by 1 projection vector to generate a P_(u) by 1 coefficient vector 430.

In the third step, illustrated with reference to FIG. 19C, the CSIcompression circuitry 320 then calculates for each of the plurality ofconfigurations a minimum squared error (MSE) fit with the P_(u) by 1coefficient vector on L evenly-spaced locations, wherein L is smallerthan N, by multiplying each of the P_(u) base sinusoids with thecorresponding coefficient in the coefficient vector 430 and taking thesummation, at each of the L evenly-spaced locations 435. The CSIcompression circuitry 320 then selects configuration u and uses itsP_(u) by 1 coefficient vector as the compressed CSI, it the total fitresidual of the MSE fit of configuration u is below a predeterminedthreshold times the minimum fit residual among all configurations, and uis such a configuration with the lowest order 440.

FIG. 20 is a flow diagram illustrating a method for compressing channelstate information (CSI) of a wireless channel, the method includes,receiving, at a receiver of a wireless communication system, anorthogonal frequency division multiplexing (OFDM) wireless signal over awireless channel 500 comprising one or more antenna pair, and measuring,at a receiver of a wireless communication system, a channel stateinformation (CSI) vector of the wireless channel from the received OFDMwireless signal for each antenna pair, wherein the CSI vector is an N by1 vector of complex numbers and wherein each complex number representsan amplitude and a phase of one of N orthogonal frequency divisionmultiplexing (OFDM) subcarriers of the wireless channel 505.

The method further includes, accessing a plurality of configurationsstored at the receiver, wherein each of the plurality of configurations“u” identifies a set of P_(u) base sinusoid vectors on constantfrequencies and wherein P_(u) is the order of the configuration which isequal to the number of complex numbers of the compressed CSI 510 ifconfiguration u is selected, calculating, for each of the plurality ofconfigurations, a dot product of the N by 1 CSI vector and a conjugateof each P_(u) base sinusoid vector identified by the selectedconfiguration to generate a P_(u) by 1 projection vector 515,calculating, for each of the plurality of configurations, a product of aconstant P_(u) by P_(u) matrix stored at the receiver and the P_(u) by 1projection vector to generate a P_(u) by 1 coefficient vector 520 andcalculating, for each of the plurality of configurations u, a minimumsquared error (MSE) fit with the P_(u) by 1 coefficient vector on Levenly-spaced locations 525, where L is smaller than N, by multiplyingeach of the P_(u) base sinusoids with the corresponding coefficient inthe coefficient vector and taking the summation, at each of the Levenly-spaced locations.

After MSE fits for all configurations at the sampling locations havebeen calculated, the method continues by selecting a configuration u anduse its P_(u) by 1 coefficient vector as the compressed CSI, if thetotal fit residual of the MSE fit of configuration u is below apredetermined threshold 530 times the minimum fit residual among allconfigurations, and u is such a configuration with the lowest order,transmitting the compressed CSI to the transmitter of the wirelesscommunication system 535, which may decompress the CSI by computing alinear combination of the base sinusoids, based upon the decompressedCSI 540.

As such, in various embodiments, the present invention describes asystem and method for compression of the CSI, referred to as CSIApx,which provides a fast and lightweight method for compressing the CSI ofOFDM wireless links. Following the proof that almost any sinusoid can beapproximated within a small bounded error as the linear combination of asmall number of base sinusoids on constant frequencies and exploitingthe constant frequencies of the base sinusoids, it is shown that CSIApxpre-computes key steps and can find a minimum square error fit of theCSI vector with very few computations.

The evaluations of CSIApx with both experimental and synthesized CSIdata, show that CSIApx achieves very good compression ratios andapproximation accuracy, significantly outperforming the existingsolution in nearly all cases at much lower computation cost. As such,CSIApx is considered to be a very useful tool to be incorporated intothe Wi-Fi protocol, and will enable timely and accurate CSI feedback toimprove the network performance.

The present invention may be implemented in hardware, because of itssimplicity and low complexity. It can also be embodied on variouscomputing platforms that perform actions responsive to software-basedinstructions. The following provides an antecedent basis for theinformation technology that may be utilized to enable the invention.

The computer readable medium described in the claims below may be acomputer readable signal medium or a computer readable storage medium. Acomputer readable storage medium may be, for example, but not limitedto, an electronic, magnetic, optical, electromagnetic, infrared, orsemiconductor system, apparatus, or device, or any suitable combinationof the foregoing. More specific examples (a non-exhaustive list) of thecomputer readable storage medium would include the following: anelectrical connection having one or more wires, a portable computerdiskette, a hard disk, a random access memory (RAM), a read-only memory(ROM), an erasable programmable read-only memory (EPROM or Flashmemory), an optical fiber, a portable compact disc read-only memory(CD-ROM), an optical storage device, a magnetic storage device, or anysuitable combination of the foregoing. In the context of this document,a computer readable storage medium may be any non-transitory, tangiblemedium that can contain, or store a program for use by or in connectionwith an instruction execution system, apparatus, or device.

A computer readable signal medium may include a propagated data signalwith computer readable program code embodied therein, for example, inbaseband or as part of a carrier wave. Such a propagated signal may takeany of a variety of forms, including, but not limited to,electro-magnetic, optical, or any suitable combination thereof. Acomputer readable signal medium may be any computer readable medium thatis not a computer readable storage medium and that can communicate,propagate, or transport a program for use by or in connection with aninstruction execution system, apparatus, or device. However, asindicated above, due to circuit statutory subject matter restrictions,claims to this invention as a software product are those embodied in anon-transitory software medium such as a computer hard drive, flash-RAM,optical disk or the like.

Program code embodied on a computer readable medium may be transmittedusing any appropriate medium, including but not limited to wireless,wire-line, optical fiber cable, radio frequency, etc., or any suitablecombination of the foregoing. Computer program code for carrying outoperations for aspects of the present invention may be written in anycombination of one or more programming languages, including an objectoriented programming language such as Java, C#, C++, Visual Basic or thelike and conventional procedural programming languages, such as the “C”programming language or similar programming languages.

Aspects of the present invention are described above with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems) and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer program instructions. These computer program instructions maybe provided to a processor of a general purpose computer, specialpurpose computer, or other programmable data processing apparatus toproduce a machine, such that the instructions, which execute via theprocessor of the computer or other programmable data processingapparatus, create means for implementing the functions/acts specified inthe flowchart and/or block diagram block or blocks.

These computer program instructions may also be stored in a computerreadable medium that can direct a computer, other programmable dataprocessing apparatus, or other devices to function in a particularmanner, such that the instructions stored in the computer readablemedium produce an article of manufacture including instructions whichimplement the function/act specified in the flowchart and/or blockdiagram block or blocks. The computer program instructions may also beloaded onto a computer, other programmable data processing apparatus, orother devices to cause a series of operational steps to be performed onthe computer, other programmable apparatus or other devices to produce acomputer implemented process such that the instructions which execute onthe computer or other programmable apparatus provide processes forimplementing the functions/acts specified in the flowchart and/or blockdiagram block or blocks.

It will be seen that the advantages set forth above, and those madeapparent from the foregoing description, are efficiently attained andsince certain changes may be made in the above construction withoutdeparting from the scope of the invention, it is intended that allmatters contained in the foregoing description or shown in theaccompanying drawings shall be interpreted as illustrative and not in alimiting sense.

It is also to be understood that the following claims are intended tocover all of the generic and specific features of the invention hereindescribed, and all statements of the scope of the invention which, as amatter of language, might be said to fall there between.

What is claimed is:
 1. A method for compressing channel stateinformation (CSI) of a wireless channel, the method comprising:receiving, at a receiver of a wireless communication system, anorthogonal frequency division multiplexing (OFDM) wireless signal over awireless channel comprising one or more antenna pair; measuring, at areceiver of a wireless communication system, a channel state information(CSI) vector of the wireless channel from the received OFDM wirelesssignal for each antenna pair, wherein the CSI vector is an N by 1 vectorof complex numbers and wherein each complex number represents anamplitude and a phase of one of N orthogonal frequency divisionmultiplexing (OFDM) subcarriers of the wireless channel; accessing aplurality of configurations stored at the receiver, wherein each of theplurality of configurations “u”, identifies a set of P_(u) base sinusoidvectors on constant frequencies and wherein P_(u) is the order of theconfiguration and is equal to the number of complex numbers of thecompressed CSI if configuration u is selected; calculating, for each ofthe plurality of configurations, a dot product of the N by 1 CSI vectorand a conjugate of each P_(u) base sinusoid vector identified by theselected configuration to generate a P_(u) by 1 projection vector;calculating, for each of the plurality of configurations, a product of aconstant P_(u) by P_(u) matrix stored at the receiver and the P_(u) by 1projection vector to generate a P_(u) by 1 coefficient vector;calculating, for each of the plurality of configurations, a minimumsquared error (MSE) fit with the P_(u) by 1 coefficient vector on Levenly-spaced locations, where L is smaller than N, by multiplying eachof the P_(u) base sinusoids with the corresponding coefficient in thecoefficient vector and taking the summation, at each of the Levenly-spaced locations; selecting configuration u and use its P_(u) by1 coefficient vector as the compressed CSI, if the total fit residual ofthe MSE fit of configuration u is below a predetermined threshold timesthe minimum fit residual among all configurations, and u is such aconfiguration with the lowest order; transmitting the compressed CSI toa transmitter of the wireless communication system; decompressing theCSI at the transmitter by computing a linear combination of the basesinusoids, using the compressed CSI as the coefficients of the basesinusoids, and adjusting the transmission characteristics of one or morewireless signals transmitted by the transmitter based upon thedecompressed CSI.
 2. The method of claim 1, further comprising, prior toaccessing a plurality of configurations stored at the receiver:computing each of a plurality base sinusoid vectors on constantfrequencies; computing a conjugate of each of the plurality of basesinusoid vectors on constant frequencies; storing each base sinusoidvector and the conjugate of each base sinusoid vector at the receiver;and storing the plurality of configurations identifying, forconfiguration u, a set of P_(u) base sinusoid vectors at the receiver.3. The method of claim 1, wherein one or of the plurality of basesinusoid vectors are shared by one or more of the plurality ofconfigurations.
 4. The method of claim 1, further comprising, prior tocalculating, for each of the plurality of configurations, computing theinverse of a constant P_(u) by P_(u) matrix and storing it at thereceiver.
 5. The method of claim 1, wherein accessing a plurality ofconfigurations stored at the receiver further comprises accessing aplurality of configurations stored at the receiver in parallel.
 6. Themethod of claim 1, further comprising, rotating the CSI vector to removethe shift frequency from the CSI vector.
 7. A wireless communicationsystem for compressing channel state information (CSI) of a wirelesschannel system, the system comprising: a receiver for receiving anorthogonal frequency division multiplexing (OFDM) wireless signal over awireless channel comprising one or more antenna pair; the receiver formeasuring a channel state information (CSI) vector of the wirelesschannel from the received OFDM wireless signal, wherein the CSI vectorfor each antenna pair is an N by 1 vector of complex numbers and whereineach complex number represents an amplitude and a phase of one of Northogonal frequency division multiplexing (OFDM) subcarriers of thewireless channel; the receiver for accessing a plurality ofconfigurations “u” stored at the receiver, identifies a set of P_(u)base sinusoid vectors on constant frequencies and wherein P_(u) is theorder of the configuration and is equal to the number of complex numbersof the compressed CSI if configuration u is selected; the receiver forcalculating, for each of the plurality of configurations, a dot productof the N by 1 CSI vector and a conjugate of each P_(u) base sinusoidvector identified by the selected configuration to generate a P_(u) by 1projection vector; the receiver for calculating, for each of theplurality of configurations, a product of a constant P_(u) by P_(u)matrix stored at the receiver and the P_(u) by 1 projection vector togenerate a P_(u) by 1 coefficient vector; the receiver for calculating,for each of the plurality of configurations, a minimum squared error(MSE) fit with the P_(u) by 1 coefficient vector on L evenly-spacedlocations, where L is smaller than N, by multiplying each of the P_(u)base sinusoids with the corresponding coefficient in the coefficientvector and taking the summation, at each of the L evenly-spacedlocations; the receiver for selecting configuration it and use its P_(u)by 1 coefficient vector as the compressed CSI, if the total fit residualof the MSE fit of configuration u is below a predetermined thresholdtimes the minimum fit residual among all configurations, and u is such aconfiguration with the lowest order; the receiver for transmitting thecompressed CSI to a transmitter of the wireless communication system;and the transmitter for decompressing the CSI by computing a linearcombination of the base sinusoids, using the compressed CSI as thecoefficients of the base sinusoids, and adjusting the transmissioncharacteristics of one or more wireless signals transmitted by thetransmitter based upon the decompressed CSI.
 8. The system of claim 7,further comprising, prior to the receiver accessing a plurality ofconfigurations stored at the receiver, the receiver computing each of aplurality base sinusoid vectors on constant frequencies, computing aconjugate of each of the plurality of base sinusoid vectors on constantfrequencies, storing each base sinusoid vector and the conjugate of eachbase sinusoid vector at the receiver and storing the plurality ofconfigurations identifying a set of base sinusoid vectors of theplurality of base sinusoid vectors at the receiver.
 9. The system ofclaim 7, wherein one or more of the plurality of base sinusoid vectorsare shared by one or more of the plurality of configurations.
 10. Thesystem of claim 7, further comprising, prior to calculating, for each ofthe plurality of configurations, computing the inverse of a constantP_(u) by P_(u) matrix and storing it at the receiver.
 11. The system ofclaim 7, wherein the receiver accessing a plurality of configurationsstored at the receiver further comprises the receiver accessing aplurality of configurations stored at the receiver in parallel.
 12. Thesystem of claim 7, further comprising, the receiver rotating the CSIvector to remove the shift frequency from the CSI vector.
 13. Anon-transitory computer-readable storage recording medium storing acomputer program used for executing a channel state information (CSI)compressing operation of an Orthogonal Frequency Division Multiplexing(OFDM) wireless channel of a wireless communication system, the computerprogram causing a wireless communication system to: receive, at areceiver of the wireless communication system, an orthogonal frequencydivision multiplexing (OFDM) wireless signal over a wireless channelcomprising one or more antenna pair; measure, at a receiver of thewireless communication system, a channel state information (CSI) vectorof the wireless channel from the received OFDM wireless signal, whereinthe CSI vector for each antenna pair is an N by 1 vector of complexnumbers and wherein each complex number represents an amplitude and aphase of one of N orthogonal frequency division multiplexing (OFDM)subcarriers of the wireless channel; access a plurality ofconfigurations stored at the receiver, wherein each of the plurality ofconfigurations “u” identifies a set of P_(u) base sinusoid vectors onconstant frequencies and wherein P_(u) is the order of the configurationand is equal to the number of complex numbers of the compressed CSI ifconfiguration u is selected; calculate, for each of the plurality ofconfigurations, a dot product of the N by 1 CSI vector and a conjugateof each P_(u) base sinusoid vector identified by the selectedconfiguration to generate a P_(u) by 1 projection vector; calculate, foreach of the plurality of configurations, a product of a constant P_(u)by P_(u) matrix stored at the receiver and the P_(u) by 1 projectionvector to generate a P_(u) by 1 coefficient vector; calculate, for eachof the plurality of configurations, a minimum squared error (MSE) fitwith the P_(u) by 1 coefficient vector on L evenly-spaced locations,where L is smaller than N, by multiplying each of the P_(u) basesinusoids with the corresponding coefficient in the coefficient vectorand take the summation, at each of the L evenly-spaced locations; selectconfiguration u and use its P_(u) by 1 coefficient vector as thecompressed CSI, if the total fit residual of the MSE fit ofconfiguration u is below a predetermined threshold times the minimum fitresidual among all configurations, and u is such a configuration withthe lowest order transmit the compressed CSI to a transmitter of thewireless communication system; decompress the CSI by computing a linearcombination of the base sinusoids, using the compressed CSI as thecoefficients of the base sinusoids, and adjusting the transmissioncharacteristics of one or more wireless signals transmitted by thetransmitter based upon the decompressed CSI.
 14. The non-transitorycomputer-readable storage recording medium of claim 13, furthercomprising, prior to accessing a plurality of configurations stored atthe receiver, the computer program causing the wireless communicationsystem to: compute each of a plurality base sinusoid vectors on constantfrequencies; compute a conjugate of each of the plurality of basesinusoid vectors on constant frequencies, store each base sinusoidvector and the conjugate of each base sinusoid vector at the receiver;and store the plurality of configurations identifying a set of basesinusoid vectors of the plurality of base sinusoid vectors at thereceiver.
 15. The non-transitory computer-readable storage recordingmedium of claim 13, wherein one or more of the plurality of basesinusoid vectors are shared by one or more of the plurality ofconfigurations.
 16. The non-transitory computer-readable storagerecording medium of claim 13, further comprising, prior the computerprogram causing the wireless communication system to calculate, for eachof the plurality of configurations, the inverse of a constant P_(u) byP_(u) matrix and store it at the receiver.
 17. The non-transitorycomputer-readable storage recording medium of claim 13, wherein thecomputer program causing the wireless communication system to access aplurality of configurations stored at the receiver further comprises thecomputer program causing the wireless communication system to access aplurality of configurations stored at the receiver in parallel.
 18. Thenon-transitory computer-readable storage recording medium of claim 13,further comprising, the computer program causing the wirelesscommunication system to preprocess the CSI vector to rotate the CSIvector to remove the shift frequency from the CSI vector.